Proof

Suppose x is a primitive element of Z/pZ and the pattern consist of the pairs (i, j) with x^{i - 1} = j mod p. Now let D' be the pattern obtained form D by a shift over the vector (a, b) and having an overlap of at least 2 with D. We will show that D = D'.

There are distinct k and l (the first coordinates of the overlapping positions) with 0 < l < k < p such that:

x^{k + a - 1} = x^{k - 1} + b mod p

and

x^{l + a - 1} = x^{l - 1} + b mod p. Hence

x^{l - 1}(x^{k - l} - 1) = x^{l + a - 1} (x^{k - l} - 1) mod p.

From this we can deduce that

x^a = 1.

However, this is only the case when a = 0. But then b is also equal to 0 and D = D'.